Surface Realization with the Intersection Segment Functional

Abstract

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry from the theoretical, algorithmic, and practical points of view.

In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection segment functional. This heuristic was used to find geometric realizations in ${\mathbb R}^3$ for all vertex-minimal triangulations of the orientable surfaces of genera $g=3$ and $g=4$. Moreover, for the first time, examples of simplicial polyhedra in ${\mathbb R}^3$ of genus 5 with 12 vertices have been obtained.